The paper by Gabe Brisson and Cliff Reiter [1] and also the text [2] describe a method
for visualizing generalizations of
the Sierpinski Triangle in any dimension. The generalization of the Sierpinski
Triangle to three dimensions is a Sierpinski Tetrahedron. This animation
shows the four dimensional
version rotated so that its structure and symmetry is more apparent. The
structure of these fractals are based on a stroke based construction. Note the
four-fold branching and the many Sierpinski Triangles appearing in planes
throughout this example.